We want to find the relationship between slope and a unit rate.
Unit Rate
A rate is a ratio that compares two quantities. Often we want to know how much of something is per 1 unit of something else. A unit rate tells us the amount of one item compared to 1 of another item. Let's look at some examples!
Scenario
Rate
Unit Rate
In a grocery store, 3 kids bought 15 candies.
15 candies per 3 kids
5 candies per 1 kid
Ana learns 20 new words every 10 minutes.
20 words per 10 minutes 10 words per 5 minute
2 words per 1 minute 120 words per 1 hour
Slope
Slope is the measure of the steepness of a line. Slope is the ratio of the rise ( vertical change) to the run ( horizontal change). Where rise corresponds to the change in y and run corresponds to the change in x.
m=rise/run=vertical change/horizontal change
As we can see, slope is a fraction. Using expansion or reduction, every fraction can be rewritten so that its denominator is 1. For example we can represent 2 as 21, - 1 as - 11, and 52 as 5/21 or 2.51.
Comparison
Slope with denominator of 1 is a rate that compares vertical change to 1 unit of horizontal change. Therefore, in proportional relationships slope is the same as a unit rate. In relationships that are not proportional slope is not defined.
